1. Technical Field
The present disclosure is generally related to the secure communication of encrypted data using quantum cryptography.
2. Description of the Related Art
Research efforts by many investigators have significantly advanced the field of quantum cryptography since the pioneering discoveries of Wiesner, Bennett and Brassard, as shown in the following references: N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. Vol. 74, pp. 145-195 (2002); S. Wiesner, “Conjugate coding,” SIGACT News Vol. 15, No. 1, pp. 78-88 (1983); C. H. Bennett and G. Brassard, “Quantum cryptography, public key distribution and coin tossing,” Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India, pp. 175-179, (IEEE 1984); C. H. Bennett and G. Brassard, “Quantum public key distribution system,” IBM Tech. Discl. Bull. Vol. 28, No. 7, pp. 3153-3163, (1985), all of which are incorporated herein by reference in their entireties. Emphasis has been placed on quantum key distribution, the generation by means of quantum mechanics of a secure random binary sequence which can be used together with the Vernam cipher (one-time pad) as discussed in G. Vernam, “Cipher printing telegraph systems for secret wire and radio telegraph communications,” J. Am. Inst. Electr. Eng. Vol. 45, pp. 295-301 (1926), which is incorporated herein by reference in its entirety, for secure encryption and decryption. Various protocols have been devised for quantum key distribution, including the single-particle four-state Bennett-Brassard protocol (BB84), Bennett (1984), the single-particle two-state Bennett protocol (B92) as in C. H. Bennett, “Quantum cryptography using any two nonorthogonal states,” Phys. Rev. Lett. Vol. 68, pp. 3121-3124 (1992), which is incorporated herein by reference in its entirety, and the two-particle entangled-state Einstein-Podolsky-Rosen (EPR) protocol as in A. K. Ekert, “Quantum cryptography based on Bell's theorem,” Phys. Rev. Lett. Vol. 57, pp. 661-663 (1991), which is incorporated herein by reference in its entirety. However the original BB84 protocol is presently perceived as the most practical and robust protocol.
One effective implementation of the BB84 protocol uses single photons linearly polarized along one of the four basis vectors of two sets of coplanar orthogonal bases oriented at an angle of 45 degrees (equivalently, π/4) relative to each other. The polarization measurement operators in one basis do not commute with those in the other, since they correspond to nonorthogonal polarization states. At a fundamental level, the potential security of the key rests on the fact that nonorthogonal photon polarization measurement operators do not commute, and this results in quantum uncertainty in the measurement of those states by an eavesdropping probe, as in H. E. Brandt, “Positive operator valued measure in quantum information processing,” Am. J. Phys. Vol. 67, pp. 434-439 (1999), which is incorporated herein by reference in its entirety. Before transmission of each photon, the transmitter and receiver each independently and randomly select one of the two bases. The transmitter sends a single photon with polarization chosen at random along one of the orthogonal basis vectors in the chosen basis. The receiver makes a polarization measurement in its chosen basis. Next, the transmitter and the receiver, using a public communication channel, openly compare their choices of basis, without disclosing the polarization states transmitted or received. Events in which the transmitter and the receiver choose different bases are ignored, while the remaining events ideally have completely correlated polarization states. The two orthogonal states in each of the bases encode binary numbers 0 and 1, and thus a sequence of photons transmitted in this manner can establish a random binary sequence shared by both the transmitter and the receiver and can then serve as the secret key, following error correction and privacy amplification, as in C. H. Bennett, G. Brassard, C. Crepeau, and V. M. Maurer, “Generalized privacy amplification,” IEEE Trans. Inf. Theor, Vol. 41, pp. 1915-1923 (1995), and C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography,” J. Cryptology, Vol. 5, pp. 3-28 (1992), both of which are incorporated herein by reference in their entireties. Privacy amplification is of course necessary because of the possibility of an eavesdropping attack, as in Gisin (2002), Bennett (1984), and Bennett (1985). Using the Vernam cipher, the key can then be used to encode a message which can be securely transmitted over an open communication line and then decoded, using the shared secret key at the receiver. (The encrypted message can be created at the transmitter by adding the key to the message and can be decrypted at the receiver by subtracting the shared secret key.)
Numerous analyses of various eavesdropping strategies have appeared in the literature, see e.g., Gisin (2002). Attack approaches include coherent collective attacks in which the eavesdropper entangles a separate probe with each transmitted photon and measures all probes together as one system, and also coherent joint attacks in which a single probe is entangled with the entire set of carrier photons. However, these approaches require maintenance of coherent superpositions of large numbers of states.